Differential Geometry

Let $\alpha (s)$ be a closed plane curve. Define the diameter $d_\alpha$ of $\alpha (s)$ to be

Now assume that the curvature $k(s) \geq 1$ for all $s$.

Let $\alpha$ be a regular $C^\infty$ curve in $\mathbb {R}^{3}$ with nonvanishing curvature. Suppose the normal vector $N(t)$ is proportional to the position vector; that is, $N(t) = c(t)\alpha (t)$ for all $t$, where $c$ is a smooth function. Determine all such curves.

A surface $S \subset \mathbb {R}^{3}$ is called triply ruled if at every $p \in S$ we can find three open line segments $L_1, L_2, L_3$ lying in $S$ such that $L_1 \cap L_2 \cap L_3 = \{ p\}$. Determine all triply ruled surfaces.

Let $\gamma : (a,b) \to \mathbb {R}^{3}$ be smooth with $\left|\gamma '\right| = 1$ and curvature $k$ and torsion $\tau$, both nonvanishing. Denote the Frenet frame by $\{ T, N, B\}$. Assume that there exists a unit vector $a \in \mathbb {R}^{3}$ with

Show that if $\gamma$ is a geodesic on the cone $z = \sqrt{x^2+y^2}$, $(x,y) \in \mathbb {R}^{2} \setminus \{ (0,0)\}$, then $\gamma$ intersects itself at most a finite number of times.

Let $\gamma : (a,b) \to \mathbb {R}^{3}$ be a $C^\infty$ curve parameterized by arc length, with curvature and torsion $k(s)$ and $\tau (s)$. Assume $k(s) \neq 0$, $\tau (s) \neq 0$ for all $s \in (a,b)$, and let $T$ and $N$ denote the unit tangent and normal vectors to $\gamma$. The curve $\gamma$ is called a Bertrand curve if there exists a regular curve $\bar\gamma : (a,b) \to \mathbb {R}^{3}$ such that for each $s \in (a,b)$ the normal lines of $\gamma$ and $\bar\gamma$ at $s$ are equal. In this case, $\bar\gamma$ is called the Bertrand mate of $\gamma$ and we can write $\bar\gamma (s) = \gamma (s) + rN(s)$ for some $r = r(s) \in \mathbb {R}$. (Note that $s$ might not be an arc length parameter for $\bar\gamma$.)

Let $x(s)$ be a curve in $\mathbb {R}^{3}$ parameterized by arc-length. Assume that $\tau (s) \neq 0$ and $k'(s) \neq 0$ for all $s$. Show that a necessary and sufficient condition for $x(s)$ to lie on a sphere is that

Let $M \subset \mathbb {R}^{3}$ be the torus obtained by rotating the circle $\{ (0, y, z) : (y-2)^2 + z^2 = 1\}$ around the $z$-axis, and let $c(t) = (2\cos t, 2 \sin t, 1)$ ("top circle"). Is this curve a geodesic on $M$? Explain without long computations.

Let $\gamma : [0,1] \to \mathbb {R}^{3}$ be a $C^\infty$ curve with $\left|\gamma '\right| = 1$ and non-vanishing curvature. Assume the torsion $\tau \equiv 0$.

Let $f : \mathbb {R} \to \mathbb {R}$ be positive and smooth. Let $M$ be the surface in $\mathbb {R}^{3}$ obtained by rotating the graph $\{ (x, f(x)) : x \in \mathbb {R}\}$ of $f$ in the $xz$ plane about the $x$ axis. Characterize in terms of $f$ the set of $x$ such that $\dfrac {1}{f(x)}$ is a principal curvature of $M$ at $(x, 0, f(x))$.

Let $\alpha (s) \subset \mathbb {R}^{3}$ be a smooth curve parameterized by arc-length. Assume that the position vector $\alpha (s)$ is always a linear combination of the binormal and normal vectors $B(s), N(s)$ of $\alpha (s)$. Show that $\alpha (s)$ does not pass through $O \in \mathbb {R}^{3}$.

Let $M^2 \subset \mathbb {R}^{3}$ be the cylinder $x^2+y^2=1$. Suppose the curve $\alpha (s) \subset M^2$ is parameterized by arclength.

Let $\Gamma$ be a two-dimensional distribution in $\mathbb {R}^{3}$ defined by

Call a normal vector field $v$ along a space curve $\gamma$ parallel if $v'$ is always tangent to $\gamma$.

Let $M$ be a surface in $\mathbb {R}^{3}$ and let $P$ be a plane. Suppose $M$ and $P$ intersect orthogonally. Show that the intersection curve (parameterized by arclength) is a geodesic on $M$.

Let $\alpha (s)$ be a $C^2$ curve in $\mathbb {R}^{3}$ parameterized by arclength. Suppose that for some function $f(s)$, $\alpha ''(s) = f(s)\alpha (s)$. What can you deduce about $f(s)$, $\alpha (s)$?

Suppose $\gamma (t)$ parameterizes a space curve with curvature function $\rho (t)$. Define a new curve $\bar\gamma$ by setting, for each $t \in \mathbb {R}$, $\bar\gamma (t) := c\gamma (t/c)$, where $c \in \mathbb {R}$ is an arbitrary fixed constant. Derive the curvature function for this new curve.

Let $\alpha : (a,b) \to \mathbb {R}^{3}$ be a smooth curve with non-vanishing curvature.

Let $\alpha$ and $\beta$ be two regular curves in $\mathbb {R}^{3}$. The curve $\beta$ is called an involute of $\alpha$ if for all $t$, $\beta (t)$ lies on the line tangent to $\alpha$ at $\alpha (t)$ and $\left\langle \alpha '(t), \beta '(t) \right\rangle = 0$. Show that every involute of a generalized helix $\alpha$ is a plane curve. (Recall that $\alpha$ is a generalized helix if for some constant vector $u \neq 0$, $\left\langle u, \alpha '(s) \right\rangle = \text{const}$, where $s$ is arclength for $\alpha$.)

Suppose that the unit normal vector to a surface $M \subset \mathbb {R}^{3}$ is constant along a regular curve $\alpha \subset M$. Deduce that in this case, $\alpha$ is an asymptotic curve, that a plane contains it, and that the Gauss curvature of $M$ vanishes at each point of $\alpha$.

[Note: A curve in a surface is called asymptotic if its acceleration is everywhere tangent to that surface.]

Sketch the closed regular plane curve $\beta : [-\pi , \pi ] \to \mathbb {R}^{2}$ having $\beta (0) = (0,0)$ and $\beta '(0) = (1,0)$, if $\beta$'s curvature function $k(s)$ ($s =$ arclength) is odd, satisfies

and has the following graph. (Include an explanation with your sketch.)

Let $X \subset \mathbb {R}^{2}$ be a connected one-dimensional real analytic submanifold, not contained in a line. Prove that not every tangent line to $X$ is bitangent -- that is, it is not the case that for all $p \in X$ there exists $q \neq p$ such that the tangent line to $X$ at $p$ equals the tangent line to $X$ at $q$ as lines in $\mathbb {R}^{2}$.

Let $M^2 \subset \mathbb {R}^{3}$ be a surface containing $x=0$. Assume that $\dfrac {\partial }{\partial x_1}$ and $\dfrac {\partial }{\partial x_2}$ are tangent to $M$ at $x=0$, and that in that basis the Weingarten map

Let $\alpha$ be the curve (near $x=0$) obtained by intersecting $M$ with the $x_1x_3$ plane.

Let $M = \{ (x,y,z) \in \mathbb {R}^{3} : z = 6 - (x^2+y^2)\}$ (a paraboloid of revolution). $D = \{ (x,y,z) \in M : z > 2\}$, and $\omega = yz^2 \, dx + xz \, dy + x^2y^2 \, dz$. Orient $M$ and evaluate

Let $S$ be a surface diffeomorphic to the ordinary 2-sphere. Suppose there is a $C^\infty$ metric of positive curvature on $S$ for which there exist two simple closed geodesics $\gamma_1$ and $\gamma_2$. Show that $\gamma_1$ and $\gamma_2$ must intersect.

Let $D$ be the disk $\{ (x,y) \in \mathbb {R}^{2} : x^2+y^2 < \tfrac 14\}$ and let $O = (0,0) \in D$. Let $(x_1,x_2) = (r,\theta )$ be the usual polar coordinates and define a metric on $D\setminus \{ O\}$ by

where $h(r,\theta ) = r^2[1 - 2r^2 + r^4\sin^2\theta ]^2$.

Let $H^2 = \{ (x,y) \in \mathbb {R}^{2}, y > 0\}$ be the upper half-plane in $\mathbb {R}^{2}$ and let $H^2$ have the Riemannian metric $g$ such that

where $(x,y)$ is the usual inner product on $\mathbb {R}^{2}$.

Consider the hyperboloid $S$ of one sheet $x^2+y^2-z^2=4$.

Calculate the total geodesic curvature of a circle of radius $r$ on a sphere of radius $R > r$.

Let $S$ be an embedded closed surface in $\mathbb {R}^{3}$ with the position vector $X(p)$ and the unit outward normal vector $N(p)$ for $p \in S$. For a fixed (small) $t$, define a surface $S_t$ to be the set $S_t = \{ X(p) + tN(p) \in \mathbb {R}^{3} \mid p \in S\}$. Let $\kappa_1, \kappa_2$ be the principal curvatures of $S$ at the point $p$ with respect to the outward normal vector. Let $H_t$ be the mean curvature of $S_t$ at the point $X(p)+tN(p)$ with respect to the outward normal vector (mean curvature is defined to be the sum of the two principal curvatures). Show that

Let $\lambda : \mathbb {R} \to \mathbb {R}$ be a smooth function with compact support, and consider the Riemannian surface obtained by equipping $\mathbb {R}^{2}$ with a metric $g$ of the form

for $\overrightarrow {v}, \overrightarrow {w} \in T_{(x,y)}\mathbb {R}^{2}$. Assuming the Gauss curvature $K$ of this metric is everywhere non-negative, use the Gauss--Bonnet Theorem to deduce that in fact, the surface is flat.

Let $M^2 \subset \mathbb {R}^{3}$ be a smooth compact surface such that $M^2 \subset \{ (x,y,z) : z \geq 0\}$. Assume that $M^2 \cap \{ (x,y,z) : z=0\}$ is a smooth curve $\alpha (s)$, parameterized by arclength.

Let $(r(s), 0, z(s))$ be a unit speed curve in $\mathbb {R}^{3}$ with $r(s) > 0$. Consider the surface of revolution $(s,\theta ) \to (r(s)\cos \theta , r(s)\sin \theta , z(s))$. On this surface compute the covariant derivatives $\nabla_{\frac{\partial }{\partial \theta }}\frac{\partial }{\partial s}$ and $\nabla_{\frac{\partial }{\partial \theta }}\frac{\partial }{\partial \theta }$ in terms of $\frac{\partial }{\partial \theta }$ and $\frac{\partial }{\partial s}$.

Let $M^2 \subset \mathbb {R}^{3}$ be an embedded compact surface of genus $\geq 1$. Show that the Gauss curvature of $M$ must vanish somewhere on $M$.

Consider the torus-of-revolution $T$ obtained by rotating the circle $(x-a)^2+z^2=r^2$ around the $z$-axis:

Parameterize this torus, compute its Gauss curvature function $K$, and verify that $\int_T K \, dA = 0$ by explicit calculation.

Let

be a curve on $S^2 \subset \mathbb {R}^{3}$. Let $X_0 = \dfrac {\partial }{\partial x_2} \in T_{\left(\frac12,0,\frac{\sqrt3}{2}\right)} S^2$. Compute the parallel translation of $X_0$ along $x(t)$.

Show that for any Riemannian metric on $S^2$ with $\left|K\right| \leq 1$, where $K$ is the Gauss curvature, the area of $S^2$ is not less than $4\pi$.

Define "geodesics", and characterize (with proof):

Give an example (e.g., draw a picture) to show that a connected surface can have two points which are not joined by any geodesic. What is the usual topological hypothesis that prevents this problem?

Define "minimal surface in $\mathbb {R}^{3}$", and prove that the catenoid, obtained by rotating the graph of $y = \cosh (x)$ around the $x$-axis in $\mathbb {R}^{3}$, is minimal.

The "Clifford" torus in $S^3$ can be parameterized using charts of the form

where $u$ and $v$ are constrained to lie within intervals length $< 2\pi$.

Let $f : \mathbb {R}^{2} \to \mathbb {R}$ be a smooth function with a critical point (e.g., a minimum or maximum) at the origin $x_1=x_2=0$.

Let $M$ be a 2-dimensional manifold smoothly embedded in $\mathbb {R}^{3}$ with unit normal $n$.

Let $M$ be a 2-dimensional manifold smoothly embedded in $\mathbb {R}^{3}$ with unit normal $n$. Assume that $M$ is the boundary of a bounded convex open set. Assume that $n$ is the exterior normal and that the Gauss curvature $K$ of $M$ is everywhere positive. [Recall that $S \subseteq \mathbb {R}^{3}$ is defined to be convex if for each two points of $S$ the line segment joining these points lies in $S$. You may use without proof the fact that $M$ lies on one side of each of its tangent planes.]

Let $M$ be a 2-dimensional manifold smoothly embedded in $\mathbb {R}^{3}$ with unit normal $n$.

By revolving the curve $\gamma$ sketched below around the $x$-axis, we get a surface of revolution $M^2 \subset \mathbb {R}^{3}$. Compute $\int_M K \, dA$, where $K$ is the Gauss curvature and $dA$ the area form, on $M$. (Make sure to justify your answer.)

A surface $\Sigma^2$ immersed in a Riemannian manifold $N$ is said to be ruled if it can be parameterized near any point by a mapping $X : (0,1)^2 \to N$ such that for each fixed $v_0 \in (0,1)$, the $u$-parameter curve $X(u,v_0)$ is a geodesic in $N$.

Let $\alpha : (0,1) \to \mathbb {R}^{3}$ be any regular arc (that is, $\alpha$ is differentiable and $\alpha '$ is nowhere zero). Let $T(u), N(u)$ and $B(u)$ be the unit tangent, normal and binormal vectors to $\alpha$ at $\alpha (u)$. Consider the normal tube of radius $\varepsilon$ around $\alpha$, that is, the surface given parametrically by $\phi (u,v) = \alpha (u) + \varepsilon \cos (v) N(u) + \varepsilon \sin (v) B(u)$.

Show that the ellipsoid $x^2+2y^2+3z^2=1$ is not isometric to any sphere $x^2+y^2+z^2=r^2$.

Let $(f(v)\cos (u), f(v)\sin (u), g(v))$ be a parametrization of a surface of revolution $S \subset \mathbb {R}^{3}$ where $(u,v) \in (0,2\pi )\times (a,b)$. If $S$ is given the induced metric from $\mathbb {R}^{3}$, prove that the following map from $S$ to $\mathbb {R}^{2}$ is locally conformal where $\mathbb {R}^{2}$ is given the standard Euclidean metric:

Let $S$ be a smooth surface in $\mathbb {R}^{3}$ defined by $r(u,v)$, where $r$ is the radius vector of $\mathbb {R}^{3}$ and $(u,v)$ are curvilinear coordinates on $S$. Let $H$ and $K$ be respectively the mean curvature and the Gaussian curvature of $S$. Let $A$ and $B$ be respectively the supremum of the absolute value of $H$ and $K$ on $S$. Let $a$ be a positive number and $n$ be the unit normal vector of $S$. Consider the surface $\tilde S$ defined by $\rho (u,v) = r(u,v) + a n(u,v)$. Let $C$ be a curve in $S$ defined by $u=u(t)$ and $v=v(t)$. Let $\tilde C$ be the curve in $\tilde S$ defined by $t \to \rho (u(t),v(t))$. Show that the length of $\tilde C$ is no less than the length of $C$ multiplied by $1-a(A+\sqrt{A^2+4B})$.

Let $u \mapsto \rho (u)$ be a smooth curve in $\mathbb {R}^{3}$. Let $S$ be the surface in $\mathbb {R}^{3}$ defined by $(u,v) \mapsto \rho (u) + v\rho '(u)$, where $\rho '(u)$ means the first-order derivative of $\rho (u)$ with respect to $u$. Assume that the two vectors $\rho '(u)$ and $\rho '(u) + v\rho ''(u)$ are $\mathbb {R}$-linearly independent at the point $(u,v) = (u_0,v_0)$, where $\rho ''(u)$ means the second-order derivative of $\rho (u)$ with respect to $u$. Verify directly from the definition of Gaussian curvature that the Gaussian curvature of $S$ is zero at the point $(u,v) = (u_0,v_0)$.

Let $M^2 \subset \mathbb {R}^{3}$ be an embedded oriented surface and let $S^2$ be the unit sphere. The Gauss map $G : M \to S^2$ is defined to be $G(x) = N(x)$ for any $x \in M$, where $N(x)$ is the unit normal vector of $M$ at $x$. Let $h$ and $g$ denote the induced Riemannian metrics on $M$ and $S^2$ from $\mathbb {R}^{3}$ respectively. Prove that if the mean curvature of $M$ is zero everywhere, then the Gauss map $G$ is a conformal map from $(M,h)$ to $(S^2,g)$. (Recall: If $(\Sigma_1,g_1)$, $(\Sigma_2,g_2)$ are two Riemannian manifolds, a map $\phi : (\Sigma_1,g_1) \to (\Sigma_2,g_2)$ is called conformal if $g_1 = \lambda \phi^*g_2$ for some scalar function $\lambda$ on $\Sigma_1$.)

Let $f : [a,b] \to \mathbb {R}^{}_+$ be smooth, and let $X \subset \mathbb {R}^{3}$ be the surface of revolution formed by revolving $z=f(x)$ about $x$-axis.

Let $\gamma$ be a geodesic curve on a regular surface of revolution $S \subset \mathbb {R}^{3}$. Let $\theta (p)$ denote the angle the curve forms with the parallel at a point $p \in \gamma$ and $r(p)$ be the distance to the axis of revolution. Prove Clairaut's relation: $r\cos \theta = \text{const}$.

Let $F(x,y,z)$ be a smooth homogenous function of degree $k$, i.e., $F(\lambda x, \lambda y, \lambda z) = \lambda^k F(x,y,z)$. Prove that away from the origin, the induced metric on the conical surface $\Sigma = \{ (x,y,z) \mid F(x,y,z) = 0\}$ has Gaussian curvature equal to $0$.

Let $X \subset \mathbb {R}^{3}$ be the cone $x^2=y^2+z^2$, and let $Y$ be the torus $(\sqrt{x^2+y^2}-2)^2+z^2=1$, that is, the torus obtained by rotating the circle $(x-2)^2+z^2-1=y=0$ around the $z$-axis.